L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s − i·27-s − 29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s − i·43-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 − 0.5i)23-s − i·27-s − 29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.255214417 + 1.160396694i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.255214417 + 1.160396694i\) |
\(L(1)\) |
\(\approx\) |
\(1.495311715 + 0.3883604780i\) |
\(L(1)\) |
\(\approx\) |
\(1.495311715 + 0.3883604780i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.85499520739882166405739857171, −26.99959778898130180713556041994, −25.70739422534221020695790978853, −25.2434136811981980230924833257, −24.192699469359491657414123515586, −23.12795372130633747172369079840, −22.056004892239609430946602316669, −20.611187128011321491989587507603, −20.13960470977256529847415152266, −18.98330966401220851292678989407, −18.05548244122135985493511609440, −17.01448031049225724399290084404, −15.41868282256486643204366413867, −14.77930531361794192609790580319, −13.55260168198889383317949756145, −12.70640352357937553350317795505, −11.56995288879767092591011334315, −9.93974348343016951960079798727, −9.08323272661118704034430934011, −7.73858665759688174745846507369, −6.98661891481067929047546245050, −5.353037384564576941856486671555, −3.73732609196479043171494260049, −2.53794783760903725117212290033, −1.0231762246715073583886723454,
1.54492013853097838910467573472, 3.16336359287306547456603572788, 4.14232418989998477138201523338, 5.64736020855343388228397059197, 7.189276553850560130118894463641, 8.43865487461016943637248963266, 9.29700987341455081320940571443, 10.43481225475748593796296511272, 11.642477211966216687033990928833, 13.04905024969734540026523115717, 14.21026093020780456973071061444, 14.764472837316160555349378674187, 16.24148425486355992660296820658, 16.77696022478718778667880185410, 18.61591540371133657769757670940, 19.224356130609751863249244484457, 20.37075900772732898371436563922, 21.312931481905460106326095387037, 22.03034848633918089080289366821, 23.4004698991376074252699861677, 24.56910490768334234090826488679, 25.34424670996564052215053967120, 26.48515617010034821733197874499, 27.05497761400928320560777513942, 28.14982250827850056901160550654